Question: Determine how many solutions exist for the system of equations. ${-4x+y = 5}$ ${8x-2y = -20}$
Solution: Convert both equations to slope-intercept form: ${-4x+y = 5}$ $-4x{+4x} + y = 5{+4x}$ $y = 5+4x$ ${y = 4x+5}$ ${8x-2y = -20}$ $8x{-8x} - 2y = -20{-8x}$ $-2y = -20-8x$ $y = 10+4x$ ${y = 4x+10}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 4x+5}$ ${y = 4x+10}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.